Borel space

Definition 0.1.

A Borel space(X;ℬ⁢(X)) is defined as a set X, together with
a Borel σ-algebra (http://planetmath.org/SigmaAlgebra) ℬ⁢(X) of subsets of X, called Borel sets. The Borel algebra on X is the smallest σ-algebra containing all open sets (or, equivalently, all closed sets if the topology on closed sets is selected).

Remark 0.1.

Borel sets were named after the French mathematician Emile Borel.

Remark 0.2.

A subspace of a Borel space (X;ℬ⁢(X)) is a subset S⊂X endowed with the relative Borel structure, that is the σ-algebra of all subsets of S of the form S⁢⋂E, where E is a Borel subset of X.